Can someone please help to understand about the concept of "lifting Hamiltonian cycles (in Cayley graphs)"?

As an example how the existence of a Hamiltonian cycle is shown by using the concept of lifting Hamiltonian cycles in the Cayley graph of $\mathbb{Z}_7 \rtimes \mathbb{Z}_3$ (A Cayley graph with respect to a generating set $S$ of the form $S=\{u,t\}$, where $|u|=7, |t|=3$). ?

I have inserted an image of a Cayley graph of the semidirect product $\mathbb{Z}_7 \rtimes \mathbb{Z}_3$ under the above mentioned type of generating set. I have drawn it so that the 7-cycles and 3-cycles are clearly visible (Vertices lablelled 1-7, 8-14, 15-21 represent 7-cycles).

Is it possible to mention, with the aid of the labelling of the vertices a corresponding Hamiltonian cycle in the quotient graph that will be lifted and how the lifting occurs?

Figure 1

Please help with this question. Thanks a lot in advance.

In the paper of the following link they have employed this concept. In 3rd page it is mentioned as Marusic's method and they are thinking about a Hamiltonian cycle in the quotient Cayley graph and extending it to a cycle in the whole graph.


Then is it possible to help me to see and understand it from the example figure I have given.

  • 1
    $\begingroup$ I think this question is a little bit unclear. What means lifting in your case? What quotient graph you are talking about here specifically? Where do you take your terminology from? Maybe some source might help. $\endgroup$
    – M. Winter
    Mar 8, 2019 at 15:51
  • $\begingroup$ Thanks a lot @M.Winter. I have edited the question now with more details, can you please help? If it's not very clear please be kind enough to tell me, I will try to include more details according to my knowledge. $\endgroup$ Mar 10, 2019 at 5:54
  • $\begingroup$ By lifting they consider the extension of the Hamiltonian cycle identified from the quotient Cayley graph to the whole graph. $\endgroup$ Mar 10, 2019 at 5:56
  • $\begingroup$ Thanks a lot again :) $\endgroup$ Mar 10, 2019 at 5:56
  • $\begingroup$ An answer from anyone is ok, please don't mind the mentioning as "Looking for an answer drawing from credible and/or official sources" $\endgroup$ Mar 12, 2019 at 9:50


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